Energy thresholds for the existence of breather solutions and travelling waves on lattices

Cuevas, J.; Karachalios, Nikos I.; Palmero, F.

doi:10.1080/00036810903277135

Cited by 12 publications

(12 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this point of view, (1.1) seems to be of particular interest due to the interplay and the expected competition of the nonlinear hopping and the generalized power nonlinearities. Extending the arguments based on variational methods [24][25][26] and the fixed point approach of [27] to establish the existence of solutions (1.8), we show the existence of lower bounds on the power of breathers on either finite or infinite lattices. The bounds depend explicitly on the dimension, and the nonlinear lattice parameters, as well as on the frequency of the solution.…”

Section: Introductionmentioning

confidence: 99%

“…It is crucial to remark that the set of parameters for which the excitation threshold R thresh is apparent suggests that the energy bounds R crit are not sharp as thresholds for existence/nonexistence. In particular, when R crit is the value derived by the fixed-point approach, it is observed that R thresh > R crit , [24,26]. For further discussions on the excitation threshold for FPU and Klein-Gordon lattices we refer the interested reader to [30].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

2012

Self Cite

View full text Add to dashboard Cite

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

2012

Self Cite

View full text Add to dashboard Cite

show abstract

“…51 ). We also remark that the results can be extended in the case of the infinite lattice Z N by implementing the concentration compactness arguments 51,53 .…”

Section: Discussionmentioning

confidence: 99%

“…If R < R thresh then I R = 0 and there is no ground state minimizer of (53). C. If R > R thresh then I R < 0 and there exists a minimizer of the variational problem (53).…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Stationary states of a nonlinear Schrödinger lattice with a harmonic trap

Achilleos

Theocharis

Kevrekidis

et al. 2011

Journal of Mathematical Physics

View full text Add to dashboard Cite

We study a discrete nonlinear Schrödinger lattice with a parabolic trapping potential. The model, describing, e.g., an array of repulsive Bose-Einstein condensate droplets confined in the wells of an optical lattice, is analytically and numerically investigated. Starting from the linear limit of the problem, we use global bifurcation theory to rigorously prove that -in the discrete regime -all linear states lead to nonlinear generalizations thereof, which assume the form of a chain of discrete dark solitons (as the density increases). The stability of the ensuing nonlinear states is studied and it is found that the ground state is stable, while the excited states feature a chain of stability/instability bands. We illustrate the mechanisms under which discreteness destabilizes the dark-soliton configurations, which become stable only inside the continuum regime. Continuation from the anti-continuum limit is also considered, and a rich bifurcation structure is revealed.

show abstract

Periodic traveling wave solutions of discrete nonlinear Klein‐Gordon lattices

Hennig

Karachalios

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

We prove the existence of periodic traveling wave solutions for general discrete nonlinear Klein‐Gordon systems, considering both cases of hard and soft on‐site potentials. In the case of hard on‐site potentials, we implement a fixed‐point theory approach, combining Schauder's fixed‐point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for nontrivial solutions to exist, energy (norm) thresholds for their existence, and upper bounds on their velocity. In the case of soft on‐site potentials, the proof of existence of periodic traveling wave solutions is facilitated by a variational approach based on the mountain pass theorem. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.

show abstract

Energy thresholds for the existence of breather solutions and travelling waves on lattices

Cited by 12 publications

References 23 publications

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

Stationary states of a nonlinear Schrödinger lattice with a harmonic trap

Periodic traveling wave solutions of discrete nonlinear Klein‐Gordon lattices

Contact Info

Product

Resources

About